acoustics and psycho acoustics - introduction to sound - part 2

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Acoustics and Psychoacoustics: Introduction tosound - Part 2

David Howard and Jamie Angus

3/5/2008 2:28 PM EST

Part 2 of an excerpt from the book "Acoustics And Psychoacoustics" covers sound intensity, power andpressure level and offers example calculations. [Part 1 discusses pressure waves and soundtransmission.]

1.2 Sound intensity, power and pressure levelThe energy of a sound wave is a measure of the amount of sound present. However, in general we are

more interested in the rate of energy transfer, instead of the total energytransferred. Therefore we are interested in the amount of energy transferred per

unit of time, that is the number of joules per second (watts) that propagate.

Sound is also a three-dimensional quantity and so a sound wave will occupyspace. Because of this it is helpful to characterise the rate of energy transfer withrespect to area, that is, in terms watts per unit area. This gives a quantity knownas the sound intensity which is a measure of the power density of a sound wavepropagating in a particular direction, as shown in Figure 1.7.

Figure 1.7 Sound intensity.

1.2.1 Sound intensity levelThe sound intensity represents the flow of energy through a unit area. In other words it represents thewatts per unit area from a sound source and this means that it can be related to the sound power level bydividing it by the radiating area of the sound source. As discussed earlier, sound intensity has a directionwhich is perpendicular to the area that the energy is flowing through, see Figure 1.7.

The sound intensity of real sound sources can vary over a range which is greater than one million-

million (1012) to one. Because of this, and because of the way we perceive the loudness of a sound, the

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sound intensity level is usually expressed on a logarithmic scale. This scale is based on the ratio of the

actual power density to a reference intensity of 1 picowatt per square metre (10-12 Wm-2).1 Thus thesound intensity level (SIL) is defined as:

SIL = 10 log10(Iactual/Iref). (1.10)

whereIactual = the actual sound power density level (in W m-2)

andIref

= the reference sound power density level (10-12 Wm-2)

The factor of 10 arises because this makes the result a number in which an integer change isapproximately equal to the smallest change that can be perceived by the human ear. A factor of 10change in the power density ratio is called the bel; in Equation 1.10 this would result in a change of 10in the outcome. The integer unit that results from Equation 1.10 is therefore called the decibel (dB). It

represents a 1010 change in the power density ratio, that is a ratio of about 1.26.

Sound power level1.2.2 Sound power levelThe sound power level is a measure of the total power radiated in all directions by a source of sound andit is often given the abbreviation SWL, or sometimes PWL. The sound power level is also expressed asthe logarithm of a ratio in decibels and can be calculated from the ratio of the actual power level to a

reference level of 1 picowatt (10-12 W) as follows:

SWL = 10 log10

(wactual

/wref

) (1.11)

where wactual = the actual sound power level (in watts)

and wref

= the reference sound power level (10-12 W)

The sound power level is useful for comparing the total acoustic power radiated by objects, for example

Example 1.6 A loudspeaker with an effective diameter of 25 cm radiates 20 mW. What is the sound

intensity level at the loudspeaker?

Sound intensity is the power per unit area. Firstly, we must work out the radiating area of theloudspeaker which is:

Aspeaker = r2 = (0.25 m/2) = 0.049 m2

Then we can work out the sound intensity as:

I= (W/Aspeaker

) = (20 x 10-3 W/0.049 m2) = 0.41 W m-2

This result can be substituted into Equation 1.12 to give the sound intensity level, which is:

SIL = 10 log10

(Iactual

/Iref

) = 10 log10

(0.41 W m-2/10-12 W m-2) = 116 dB

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ones which generate unwanted noises. It has the advantage of not depending on the acoustic context, aswe shall see in Chapter 6. Note that, unlike the sound intensity, the sound power has no particulardirection.

1.2.3 Sound pressure level

The sound intensity is one way of measuring and describing the amplitude of a sound wave at aparticular point. However, although it is useful theoretically, and can be measured, it is not the usualquantity used when describing the amplitude of a sound. Other measures could be either the amplitudeof the pressure, or the associated velocity component of the sound wave.

Because human ears are sensitive to pressure, which will be described in Chapter 2, and because it iseasier to measure, pressure is used as a measure of the amplitude of the sound wave. This gives aquantity which is known as the sound pressure, which is the root mean square (rms) pressure of a soundwave at a particular point. The sound pressure for real sound sources can vary from less than 20

micropascals (20 Pa or 20 - 10-6 Pa) to greater than 20 pascals (20 Pa).2

Note that 1 Pa equals a pressure of 1 newton per square metre (1 N m

-2

). These two pressures broadlycorrespond to the threshold of hearing (20 Pa) and the threshold of pain (20 Pa) for a human being, at afrequency of 1 kHz, respectively. Thus real sounds can vary over a range of pressure amplitudes whichis greater than a million to one. Because of this, and because of the way we perceive sound, the soundpressure level is also usually expressed on a logarithmic scale. This scale is based on the ratio of theactual sound pressure to the notional threshold of hearing at 1 kHz of 20 Pa. Thus the sound pressurelevel (SPL) is defined as:

SPL = 20 log10

(pactual

/pref

) (1.12)

wherepactual = the actual pressure level (in Pa)

andpref= the reference pressure level (20 Pa)

The multiplier of 20 has a twofold purpose. The first is to make the result a number in which an integerchange is approximately equal to the smallest change that can be perceived by the human ear. Thesecond is to provide some equivalence to intensity measures of sound level as follows.

Sound pressure level (cont.)

The intensity of an acoustic wave is given by the product of the volume velocity 3 and pressure

Example 1.7 Calculate the SWL for a source which radiates a total of 1 watt.

Substituting into Equation 1.11 gives:

SWL = 10 log10(wactual/wref) = 10 log10(1 watt/1 x 10-12 W)

= 10 log10

(1 x 1012) = 120 dB

A sound pressure level of one watt would be a very loud sound, if you were to receive all the power.However, in most situations the listener would only be subjected to a small proportion of this power.

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amplitude:

Iacoustic

= Up

wherep = the pressure component amplitudeand U= the volume velocity component amplitude

However, the pressure and velocity component amplitudes are linked via the acoustic impedance(Equation 1.9) so the intensity can be calculated in terms of just the sound pressure and acousticimpedance by:

Iacoustic

= Up = (p/Zacoustic

)p =p2/Zacoustic

Therefore the sound intensity level could be calculated using the pressure component amplitude and theacoustic impedance using:

SIL = 10log10(Iacoustic/Iref) = 10 log10(p2/(Zacoustic/Iref)) = 10 log10(p

2/(ZacousticIref))

This shows that the sound intensity is proportional to the square of the pressure, in the same way thatelectrical power is proportional to the square of the voltage. The operation of squaring the pressure canbe converted into multiplication of the logarithm by a factor of two, which gives:

SIL = 20 log10

(p/(Zacoustic

Iref

))

This equation is similar to Equation 1.12 except that the reference level is expressed differently. In fact,this equation shows that if the pressure reference level was calculated as:

pref= ZacousticIref= (416 x 10-12

) = 20.4 x 10-6

(Pa)

then the two ratios would be equivalent. The actual pressure reference level of 20 Pa is close enough tosay that the two measures of sound level are broadly equivalent. That is, SILSPL for a single soundwave a reasonable distance from the source and any boundaries. They can be equivalent because thesound pressure level is calculated at a single point and sound intensity is the power density from a soundsource at the measurement point. However, whereas the sound intensity level is the power density froma sound source at the measurement point, the sound pressure level is the sum of the sound pressurewaves at the measurement point.

If there is only a single pressure wave from the sound source at the measurement point, that is there are

no extra pressure waves due to reflections, the sound pressure level and the sound intensity level areapproximately equivalent, SILSPL. This will be the case for sound waves in the atmosphere wellaway from any reflecting surfaces. It will not be true when there are additional pressure waves due toreflections, as might arise in any room or if the acoustic impedance changes. However, changes in levelfor both SIL and SPL will be the equivalent because if the sound intensity increases then the soundpressure at a point will also increase by the same proportion. This will be true so long as nothing altersthe number and proportions of the sound pressure waves arriving at the point at which the soundpressure is measured. Thus, a 10 dB change in SIL will result in a 10 dB change in SPL.

These different means of describing and measuring sound amplitudes can be confusing and one must be

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careful to ascertain which one is being used in a given context. In general a reference to sound levelimplies that the SPL is being used because the pressure component can be measured easily andcorresponds most closely to what we hear.

Let us calculate the SPLs for a variety of pressure levels.

Coming up in Part 3: Adding sounds together.

Footnotes:1. The symbol for power in watts is W.

2. The pascal (Pa) is a measure of pressure; 1 pascal (1 Pa) is equal to 1 newton persquare metre (1 Nm -

2).

3. Volume velocity is a measure of the velocity component of the wave. It is measured in units of litresper second (ls-1).

Printed with permission from Focal Press, a division ofElsevier. Copyright 2006. "Acoustics andPsychoacoustics" by David Howard and Jamie Angus. For more information about this title, please visitwww.focalpress.com.

Related links:Audio in the 21st Century - Sound

Example 1.8 Calculate the SPL for sound waves with rms pressure amplitudes of 1 Pa, 2 Pa and 2 Pa.

Substituting the above values of pressure into Equation 1.12 gives:

SPL1 Pa = 20 log10(pactual/pref) = 20 log10(1 Pa/20 Pa)

= 20 - log10

(5 x 104) = 94 dB

1 Pa is often used as a standard level for specifying microphone sensitivity and, as the abovecalculation shows, represents a loud sound.

SPL2 Pa = 20 log10(pactual/pref) = 20 log10(2 Pa/20 Pa)

= 20 - log10

(1 x 105) = 100 dB

Doubling the pressure level results in a 6 dB increase in sound pressure level, and a tenfold increase inpressure level results in a 20 dB increase in SPL.

SPL2 Pa

= 20 log10

(pactual

/pref

) = 20 log10

(2 Pa/20 Pa)

= 20 - log10

(1 x 10-1) = -20 dB

If the actual level is less than the reference level then the result is a negative SPL. The decibel conceptcan also be applied to both sound intensity and the sound power of a source.

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acoustics and psycho acoustics - introduction to sound - part 2

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